On non-negative operators in Krein spaces and their perturbations

TL;DR

This paper introduces a new spectral characterization for non-negative operators in Krein spaces, defining local non-negativity and analyzing their perturbations.

Contribution

It provides a novel spectral and local sign property-based characterization of non-negative operators in Krein spaces and explores their perturbations.

Findings

New spectral characterization based on local sign properties.

Definition of local non-negativity for self-adjoint operators.

Identification of classes of operators as perturbations of non-negative operators.

Abstract

One of the most important contributions of Heinz Langer in the area of operator theory in Krein spaces is the introduction of the notion of definitizable operators and the construction of the corresponding spectral function. In this note we obtain a new characterization for the subclass of non-negative operators in Krein spaces which is based on local sign type properties of the spectrum and growth conditions on the resolvent. Based on these local properties, a notion of local non-negativity for self-adjoint operators in Krein spaces is defined and it is shown that such classes of operators appear naturally as perturbations of non-negative operators.